Video Transcript
Determine the negative integral of csc squared two 𝑥 with respect to 𝑥.
We begin by recalling the standard result of the indefinite integral of the square of the cosecant function. The integral of csc squared 𝑥 with respect to 𝑥 is equal to negative cot 𝑥 plus 𝐶. We can extend this using the laws of integration to recall that the integral of csc squared 𝑎𝑥 with respect to 𝑥 is equal to negative one over 𝑎 cot 𝑎𝑥 plus 𝐶.
In this question, our value of 𝑎 is two. This means that the integral of csc squared two 𝑥 with respect to 𝑥 is equal to negative one-half cot two 𝑥 plus 𝐶. As we want the negative of this, our answer is as shown. Simplifying our answer, we see that the negative integral of csc squared two 𝑥 with respect to 𝑥 is equal to one-half cot two 𝑥 plus 𝐶.
